A two-step sequential procedure for detecting an epidemic change

The typical approach in change-point theory is to perform the statistical analysis based on a sample of fixed size. Alternatively, and this is our approach, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the "normal" behaviour. In this paper we focus on epidemic changes, that is, a first change (the outbreak) when there is a change in the distribution, and a second change, when the process regains its ordinary structure. Based on the counting process related to the original process observed at equidistant time points, we propose some stopping rules for this to happen and consider their asymptotics under the null hypothesis as well as under alternatives. The main basis for the proofs are strong invariance principles for renewal processes, extreme value asymptotics for Gaussian processes, and the law of the iterated logarithm.